> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/260-function-theory/topics/function-in-mathematics.md). # Function in mathematics A **function** (map, mapping, correspondence, transformation) is a binary relation between two sets that associates to each element of the first set exactly one element of the second set. A function is a process (rule) that associates each element x of a set X (the domain of the function) to a single element y of a set Y (the codomain of the function), with the possibility that the domain and codomain are the same set. Functions originally represented how a varying quantity depends on another quantity; e.g. the position of a planet is a function of time. The concept of a function was formalized at the end of the 19th century in terms of set theory. A function is a relation that is right-unique and left-serial. This relation is denoted by `f (x) = y`, where the element `x` is the input (argument) value to `f` and `y` is its output value; it is also said that `y` is the image of `x` by `f`, with `x` as the pre-image. The input \*argument, `M`, is bound by a function, `f`, by a corresponding *formal parameter*, `x`, which is its *declaration occurrence*, while all the places inside the function's body, where that parameter is used, are called its *application occurrences*. ``` ┌ function application ``` ┌ function ┐ │ to an │declaration│ │ ┌ argument │ │ │ │ f(x) = x + 2x f(5) = 5 + 2\*5 │ │ │ │ parameter │ │ │ │ │ └────┴─ application (x+2x)\[5/x] │ │ │ │ occurrences substitution │ │ │ └ parameter declaration occurrence │ └ name of the function A function, just like a relation, is uniquely represented by a set of ordered pairs, called the *graph of the function*, `f = { ∀x ∈ X. ∃y ∈ Y | (x, y) }`. `∀x ∈ X. ∃y ∈ Y. (x, y) ∈ f ∧ y = f(x)`